Sur les Formules Explicites I: analyse invariante

Weil has generalized the Riemann-von Mangoldt explicit formula linking the prime numbers with the zeros of the zeta function to the set-up of a general algebraic number field K and Dirichlet-Hecke L-function, revealing in the process the role played by the completions (finite and infinite) of K. We show how the local terms of these explicit formulae are explained by the dilaton invariant ``conductor operator'' log(|x|) + log(|y|). We also check Weil's positivity criterion under a support condition.